91Ó°ÊÓ

When dealing with second-order linear differential equations like the one given, identifying the characteristic equation is the first step to finding its solution. This part involves assuming a solution of the form \( y = e^{rt} \), which is typical for homogeneous equations. By substituting \( y \), \( y' \), and \( y'' \) into the differential equation, we can extract a polynomial equation in \( r \). This equation is known as the characteristic equation. For the given example, the differential equation is \( y'' + 10y' + 25y = 0 \). By substituting the assumed solution, we derive the characteristic equation:

• \( r^2 + 10r + 25 = 0 \)

This quadratic equation reflects how the behavior of the differential equation is governed by the roots of this characteristic polynomial.

Once the characteristic equation \( r^2 + 10r + 25 = 0 \) is determined, the next step is solving it. As it is a quadratic equation, we use the quadratic formula: \[ r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] where \( a \), \( b \), and \( c \) are coefficients from the characteristic equation. In this specific case, \( a = 1 \), \( b = 10 \), and \( c = 25 \). By substituting these values, we find: \[ r = \frac{-10 \pm \sqrt{10^2 - 4 \cdot 1 \cdot 25}}{2 \cdot 1} \] which simplifies to \( r = \frac{-10 \pm 0}{2} \), yielding \( r = -5 \). This computation shows us the roots that determine the specific solution to the differential equation.

In differential equations, when solving the characteristic equation, we may encounter repeated roots. A repeated root occurs when the discriminant \( b^2 - 4ac \) equals zero, resulting in the same root occurring twice. This is what happens in our example, where the root \( r = -5 \) appears twice. Repeated roots present a unique situation. Instead of a simple exponential solution, the general solution requires an additional modification. So, for repeated roots, the general solution includes both the root and a term that increases linearly: \[ y(t) = (C_1 + C_2t)e^{-5t} \] Here, \( C_1 \) and \( C_2 \) are arbitrary constants, which will be determined if initial conditions are provided.

The term 'homogeneous' in the context of differential equations like \( y'' + 10y' + 25y = 0 \) implies that all terms in the equation depend solely on the variable \( y \) and its derivatives, without any external function or constants. Homogeneous equations often lead to characteristic equations, which greatly simplify solving the differential equation using roots of the resulting polynomial.

• These roots inform us about the behavior of solutions, be it real and distinct, repeated, or complex.
• The nature and number of these roots directly manipulate the form of the solution, whether purely exponential, a combination of exponentials and polynomials (as in repeated roots), or involving sines and cosines if the roots are complex.

The process of solving these equations helps in understanding more intricate behaviors described by the differential equations.